Continuous Function Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
Apply the Intermediate Value Theorem to show that h(x)=x3−x−1h(x) = x^3 - x - 1 has a root in the interval [1,2][1, 2].

Solution

  1. 1
    Compute endpoint values: h(1)=1−1−1=−1<0h(1) = 1-1-1 = -1 < 0 and h(2)=8−2−1=5>0h(2) = 8-2-1 = 5 > 0.
  2. 2
    Since hh is a polynomial (continuous everywhere) and h(1)<0<h(2)h(1) < 0 < h(2), by the IVT there exists c∈(1,2)c \in (1,2) such that h(c)=0h(c) = 0.

Answer

By IVT, hh has a root in (1,2)(1, 2)
The Intermediate Value Theorem guarantees that a continuous function takes every value between its endpoint values. Since hh changes sign on [1,2][1,2], it must cross zero somewhere in between.

About Continuous Function

A function is continuous at a point if the limit equals the function value there, with no jumps, holes, or vertical asymptotes in the interval of interest.

Learn more about Continuous Function →

More Continuous Function Examples

Example 1 easy

Show that [formula] is continuous at [formula] using the three-part definition of continuity.

Example 2 hard

Find where [formula] is discontinuous, classify the discontinuity, and determine if it can be remove

Example 3 easy

Is [formula] continuous on [formula]? If not, identify where and why.

← All Continuous Function Examples Continuous Function Concept